Problem 13 - Essential Calculus

Question

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \sqrt{n}}{n+2} $$

Step-by-Step Solution

Answer

By applying the Alternating Series Test, it has been determined that the given series converges.

Step 1: Checking that the terms of the sequence are decreasing

The sequence in question is \(b_n = \frac{\sqrt{n}}{n+2}\). To prove that \(b_{n}\) decreases as \(n\) increases, it is enough to show that \(b_{n+1} < b_{n}\) for all \(n\). So let's calculate \(b_{n+1}\) and \(b_{n}\). Therefore, check the inequality: \(b_{n+1} = \frac{\sqrt{n+1}}{n+3} < \frac{\sqrt{n}}{n+2} = b_{n}\). This inequality is true for all \(n \geq 1\), so condition (1) is satisfied.

Step 2: Checking the limit of the sequence is zero

The second condition for the Alternating Series Test is to prove that \( \lim_{{n}\to{\infty}} b_n = 0 \) . The calculation is \( \lim_{{n}\to{\infty}} \frac{\sqrt{n}}{n+2} = 0 \), which satisfies condition (2).

Step 3: Concluding the convergence of the series

As both conditions are met for the Alternating Series Test (Leibniz's Test), the given series is convergent.

Key Concepts

Series Convergence

Series convergence is a fascinating topic in calculus and analysis. When we talk about the convergence of a series, we mean that the sum of its infinite terms results in a finite number. The convergence or divergence of a series can be understood using various tests. One such method is the

• Alternating Series Test: This test is particularly handy for series where the terms alternate in sign, i.e., the series takes both positive and negative values in turn.
• Requirements: For an alternating series to converge, the absolute value of the terms must decrease over time and tend toward zero.

In our exercise, we used the Alternating Series Test to analyze convergence. Since the terms of the series defined by \(rac{(-1)^{n+1} \sqrt{n}}{n+2}\)meet the conditions of decreasing in magnitude and having a limit of zero, the series is concluded to be convergent.

Sequence Limits

The concept of sequence limits is crucial when exploring series. A sequence is essentially an ordered list of numbers generated by some rule. Determining the limit of a sequence involves figuring out what value the terms of a sequence approach as they keep extending to infinity.For sequences in an alternating series:

• The limit needs to be zero to consider convergence; otherwise, the series may diverge or behave unpredictably. This forms part of the conditions checked in the Alternating Series Test.
• In our example, the sequence given as \( b_n = rac{\sqrt{n}}{n+2} \) was shown to have a limit tending toward zero. As \(n\) becomes larger, the function approaches zero, satisfying the second condition for convergence.

Understanding sequence limits helps in evaluating terms' behavior over time and is pivotal in determining the overall nature of an infinite series.

Decreasing Sequence

A key criterion in the Alternating Series Test is that the series terms must be a decreasing sequence in terms of absolute value. A sequence is deemed decreasing if each term is less than or equal to the preceding term.To show this:

• We typically compare two consecutive terms, \(b_n\) and \(b_{n+1}\), proving \(b_{n+1} < b_n\). This inequality establishes that the sequence is decreasing.
• In our problem, the sequence \( b_n = \frac{\sqrt{n}}{n+2}\) was shown to decrease as \(n\) rises, through the relationship \(\frac{\sqrt{n+1}}{n+3} < \frac{\sqrt{n}}{n+2}\).

Each term in the sequence being smaller than the previous one is crucial in both proving convergence through the Alternating Series Test and ensuring the series behaves predictably. Without a decreasing sequence, a series may not provide the predictable sum we're aiming to define.